SANA, Yemen — More than three weeks after Saudi Arabia began a bombing campaign aimed at crippling his movement, the leader of Yemen’s Houthi rebels responded with defiance on Sunday in a televised speech, saying that Saudi attempts to “humiliate” his country were doomed.

“Those who want the people to give in are just dreaming,” the leader, Abdel-Malik al-Houthi, said in a long, fuming address that was also sharply critical of the United States. “Our Yemeni people have the right to fend off the aggression, and to confront the aggressor with all possible and available means,” he said on a Houthi news channel.

Mr. Houthi gave no indication that he was willing to negotiate with the Saudis or to agree to their conditions for a cease-fire, including that the Houthis withdraw from cities they have captured. His defiance suggested a prolonged war and raised further questions about the Saudi government’s military strategy, which has been centered on pounding the Houthis and their allies from the air.

The Saudis launched the intervention with backing from the United States and a coalition of Arab states, aiming to restore Yemen’s exiled government, led by President Abdu Rabbu Mansour Hadi. The coalition’s warplanes have carried out hundreds of airstrikes, targeting military bases and positions held by the Houthis and their main allies, military units and security forces loyal to Ali Abdullah Saleh, who ceded the presidency in 2012.

The intensity of the Saudi bombing has left Yemen facing the likelihood of a devastated military, even when this conflict ends. The airstrikes have also hit factories and homes, killing dozens of civilians and arousing misgivings even among Saudi allies.

Residents in Sana, the capital, have been subjected to daily airstrikes for weeks. And in the southern port city of Aden, hundreds of people have been killed in street fighting between the Houthis and Mr. Saleh’s forces on one side, and local militias representing Yemen’s southern separatists on the other.

In the latest sign the war is spreading, dozens of people have been killed in recent days in Taiz, Yemen’s third largest city, where fighters loyal to Mr. Hadi, backed by Saudi airstrikes, are battling the Houthis and their allies.

Several military units have switched sides by declaring their loyalty to the exiled president, Mr. Hadi. On Sunday, Yemeni officials told the Reuters news agency that a brigadier general who commands a district along the Saudi border and leads 15,000 troops had also pledged his support for Mr. Hadi.

It remained to be seen whether the defections would tip the balance in the conflict or simply further fracture the country. A multiplying cast of combatants are being drawn into the fight — including military units with shifting loyalties, armed tribesmen, separatist fighters and extremist militants with Al Qaeda’s Yemeni branch — and are confusing the battle lines.